Critical Assessment of Models for Transport of ... - ACS Publications

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Critical Assessment of Models for Transport of Engineered Nanoparticles in Saturated Porous Media Eli Goldberg,† Martin Scheringer,*,† Thomas D. Bucheli,‡ and Konrad Hungerbühler† †

Institute for Chemical and Bioengineering, ETH Zürich, 8093 Zürich, Switzerland Agroscope Institute for Sustainability Sciences ISS, 8046 Zürich, Switzerland

‡

S Supporting Information *

ABSTRACT: To reliably assess the fate of engineered nanoparticles (ENP) in soil, it is important to understand the performance of models employed to predict vertical ENP transport. We assess the ability of seven routinely employed particle transport models (PTMs) to simulate hyperexponential (HE), nonmonotonic (NM), linearly decreasing (LD), and monotonically increasing (MI) retention profiles (RPs) and the corresponding breakthrough curves (BTCs) from soil column experiments with ENPs. Several important observations are noted. First, more complex PTMs do not necessarily perform better than simpler PTMs. To avoid applying overparameterized PTMs, multiple PTMs should be applied and the best model selected. Second, application of the selected models to simulate NM and MI profiles results in poor model performance. Third, the selected models can well-approximate LD profiles. However, because the models cannot explicitly generate LD retention, these models have low predictive power to simulate the behavior of ENPs that present LD profiles. Fourth, a term for blocking can often be accounted for by parameter variation in models that do not explicitly include a term for blocking. We recommend that model performance be analyzed for RPs and BTCs separately; simultaneous fitting to the RP and BTC should be performed only under conditions where sufficient parameter validation is possible to justify the selection of a particular model.

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principles of particle transport modeling are valid for ENPs.12 In general, ENPs are prone to aggregate and preferentially deposit in the column inlet and produce hyperexponential retention profiles (RPs).13−16 However, ENPs also exhibit linearly decreasing (LD), nonmonotonic (NM), or monotonically increasing (MI) RPs.17−20 It is important to note that current PTMs are neither mathematically, nor conceptually, capable of generating LD, NM, and MI RPs (but several of current PTMs are able to generate HE RPs). This indicates that the models do not capture the processes that lead to LD, NM or MI RPs and are mechanistically not adequate for these types of RPs. Interestingly, although LD, NM, and MI RPs cannot be described by the current PTMs, the models often yield breakthrough curves (BTCs) that are in close agreement with the BTCs measured in combination with LD, NM, and MI RPs. In this situation, the PTMs are often presumed to provide predictive insights to the behavior, and therefore the fate, of nanoparticles.13,16,19 The models are routinely fitted to measured RP and BTC data and parameters, such as deposition rate constants, remobilization rate constants, and maximum adsorption capacity, are routinely derived from the fit and

INTRODUCTION Environmental applications of nanomaterial-based products, particularly engineered nanoparticle (ENP) plant protection products (PPPs), represent a small, but rapidly growing, portion of the global nanotechnology market.1 The flux of nano titanium dioxide, nanosilver, and carbon nanotubes from direct application of ENP pesticides and PPPs to soil could be between 1 μg kg−1 a−1 and 11 mg kg−1 a−1.1 The detection of ENPs in complex environmental matrices is not yet analytically feasible.2,3 Therefore, ENP fate assessment in the subsurface must, for now, depend on model-derived determinations of ENP transport from soil column experiments. Particle transport models based on the advection-dispersion equation (herein referred to as PTMs) are often employed to describe the transport of particles through saturated porous media. Particle deposition and remobilization, or the transfer of particles between fluid and stationary phases, are described as first-order processes in these models. More complex PTMs additionally include parameters that influence deposition processes to consider heterogeneity in particle attachment efficiencies, blocking effects, multisite kinetics, and/or pore structure influences (i.e., depth-dependent retention).4−8 Numerous PTMs have been developed to predict the transport of colloids in saturated porous media.4−6,9−11 However, the applicability of these PTMs to ENPs has not been thoroughly examined. ENPs are viewed as a subclass of colloids ranging from 1−100 nm and it is thought that the basic © 2014 American Chemical Society

Received: Revised: Accepted: Published: 12732

April 25, 2014 September 24, 2014 September 26, 2014 September 26, 2014 dx.doi.org/10.1021/es502044k | Environ. Sci. Technol. 2014, 48, 12732−12741

Environmental Science & Technology

Article

Table 1. Particle Transport Models Investigated in This Studya model description

modeled process(es)

governing mass balances

free parameters

refs

M1

colloid filtration theory (CFT)

deposition

ρ ∂S ∂C ∂C = − νp − b ∂t ∂x θ ∂t ρb ∂S 3(1 − θ) = kdC ; kd = ανpη0 θ ∂t 2dc

α

9,27

M2

dual deposition mode CFT

fast deposition

ρ ∂S ∂C ∂C = − νp − b ∂t ∂x θ ∂t ρb ∂S = fkd,f C + (1 − f )kd,sC θ ∂t

f

4,10,29

slow deposition

kd,f =

M3

single-site deposition remobilization

dual-site deposition remobilization

kd

ρ ∂S ρ ∂S ∂C ∂C ∂ 2C = Dp 2 − νp − b − b 2 ∂t ∂x θ ∂t θ ∂t ∂x ρb ∂S ρb = kdC − k rS θ ∂t θ ρb ∂S2 ρ = kd,2C − k r,2S2 θ ∂t θ

kd kd,2

deposition

ρ ∂S ∂C ∂ 2C ∂C = Dp 2 − νp − b ∂t ∂x θ ∂t ∂x

kd

remobilization Langmuirian blocking

ρb ∂S

deposition

ρ ∂S ∂C ∂ 2C ∂C = Dp 2 − νp − b ∂t ∂x θ ∂t ∂x ρb ∂S ρ = kdψsC − k rS θ θ ∂t

kd

depth-dependent retention

⎛ d + x ⎞−β ψs = ⎜ c ⎟ ⎝ dc ⎠

β

deposition

ρ ∂S ∂C ∂ 2C ∂C = Dp 2 − νp − b ∂t ∂x θ ∂t ∂x ρb ∂S ρ = kdψbψsC − k rS θ ∂t θ

kd

deposition remobilization

M5

M6

single-site deposition remobilization and blocking

single-site deposition remobilization and depthdependent retention

remobilization

M7

single-site deposition remobilization, blocking, and depth-dependent retention

3(1 − θ) 3(1 − θ) αf νpη0 ; kd,s = αsνpη0 2dc 2dc

ρ ∂S ∂C ∂ 2C ∂C = Dp 2 − νp − b ∂t ∂x θ ∂t ∂x ρb ∂S ρb = kdC − k rS θ ∂t θ

deposition remobilization

M4

αs

remobilization Langmuirian blocking depth-dependent retention

θ ∂t

= kdψbC −

⎛ ρ S ⎞ k rS ; ψb = ⎜1 − ⎟ θ Sm ⎠ ⎝

⎛ d + x ⎞−β ⎛ S⎞ ψb = ⎜1 − ⎟; ψs = ⎜ c ⎟ Sr ⎠ ⎝ ⎝ dc ⎠

28

kr

6

kr kr,2

5,16

kr Sm

5,7,8,30

kr

7,16,30

kr Sm β

Legend: C: aqueous ENP concentration [kg m−3]; S2: 2nd site solid-phase concentration [kgENPs kgsoil−1]; vp: pore water velocity [m s−1]; t: time [s]; αf, αs: fast/slow attachment efficiency [−]; x: distance from column inlet [m] η0: single-collector contact efficiency [−]; ρb: media bulk density [kg m−3]; Dp: longitudinal dispersion coefficient [m2s−1]; θ: granular media porosity [−]; S: primary site solid-phase concentration [kgENPs kgsoil−1]; kd,2: secondary site deposition rate constant [s−1]; dc: granular media average diameter [m]; kr: primary site remobilization rate constant [s−1]; α: attachment efficiency [−]; kr,2: secondary site remobilization rate constant [s−1]; kd: primary site deposition rate constant [s−1]; ψb: dynamic Langmuirian blocking function [−]; Sm: maximum solid-phase concentration [kgENPs kgsoil−1]; f: particle population fraction associated with αf [−]; ψs: depth-dependent retention function [−]; kd,f: rate constant of deposition associated with αf [s−1]; β empirical depth-dependent retention parameter [−]; kd,s: rate constant of deposition associated with αs [s−1] a

reported.15,17−19,21,22 However, for cases where the models are mechanistically inadequate, good agreement between modeled

and measured BTCs only indicates that the models may be used as a means of describing the observed breakthrough. It is 12733

dx.doi.org/10.1021/es502044k | Environ. Sci. Technol. 2014, 48, 12732−12741


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where C is the fluid phase particle concentration, t is time, x is distance from the column inlet, Dp is the longitudinal dispersion coefficient, vp is the average pore velocity, θ is the porosity, which is equivalent to the volumetric water content at complete saturation, and ρb is the media bulk density.26,27 The solid-phase mass balance is

unknown whether this descriptive ability reaches beyond the narrow experimental range under which the model was fitted to the measured data. Therefore, there is a need to differentiate between cases where some of the PTMs may have some predictive power (HE RPs), cases where some of the PTMs are, at best, useful as descriptive tools for BTCs (LD and NM RPs), and cases where the models are fundamentally inadequate for the data from a soil-column experiment and should not be applied at all (MI RPs). Moreover, there is a need for guidance on how model parameters derived from a fit to measured RP and/or BTC data should be interpreted. In general, model fit alone does not provide a sufficient basis for inferring process descriptions for mechanisms within soil columns.23,24 More specifically, several factors contribute to a general misunderstanding of the capabilities of current PTMs to infer mechanistic understanding of ENP transport. First, validation of fitted parameters is often impossible, as some model parameters have no well-defined physical meaning or because data sets are relatively small in comparison to the number of fitted parameters.25 Second, challenges often arise in the parameter fitting procedure itself, as collectively fitting free parameters by optimizing the agreement between model and measured data may yield large parameter uncertainties and, in some cases, nonunique parameter solutions.25 Finally, it has been reported that multiple mechanistically different models fit measurements of ENP retention with similar goodness-of-fit, which undermines the ability of these models to provide mechanistic insights.16,21 These factors do not necessarily preclude useful application of some of the PTMs in certain cases. However, they need to be addressed more explicitly in order to avoid misleading or incorrect application of PTMs to ENP transport. The purpose of this work is not to claim that any of these models are “realistic”, or particularly suitable for modeling the movement of ENPs in soil. Our intention is to evaluate the ability of these models to describe the presented retention profiles and breakthrough curves because these models are used frequently by scientists in the field. Through this work we aim to (i) to reduce confusion concerning the suitability of PTMs for describing ENP transport in the subsurface by increasing awareness of model limitations and (ii) to improve on current environmental fate assessment practices for ENPs. In support of these goals, we systematically evaluate the suitability of seven routinely employed PTMs to describe representative HE, LD, NM, and MI retention profiles from published soil-column experiment data. Each of the seven examined PTMs is fitted to each retention profile. For all models, the resulting breakthrough curves are generated from the parameters determined by fitting the RP and visually evaluated against a simultaneous fit of the RP and BTCs from the publication from which the presented profile is obtained. The suitability and performance of each PTM is then visually and statistically evaluated.

ρb ∂S θ ∂t

(2)

where S is the solid-phase particle concentration, kd is the firstorder deposition rate constant, and kr is the first-order remobilization rate constant.27,28 Equations 1 and 2 form the fundamental single-site deposition-remobilization PTM, presented as M3, from which six model modifications are derived, see Table 1. The seven PTMs are presented in order of increasing complexity as follows: a colloid filtration theory (CFT) model (M1), a CFT model that includes a bimodal distribution of particle attachment efficiencies (M2), a single-site depositionremobilization model (M3), a dual-site deposition-remobilization model (M4), a single-site deposition-remobilization model that includes dynamic Langmuirian blocking (M5), a single-site deposition-remobilization model that includes depth-dependent retention (M6), and a single-site deposition-remobilization model that includes both dynamic Langmuirian blocking and depth-dependent retention (M7). For each model, Table 1 presents the modeled processes, governing equations, indicates the fitted parameters, and lists key references. Model setup, shared assumptions, boundary conditions, and programming details for the considered PTMs are provided in the Supporting Information (SI). Nanospecific Model Considerations. It is important to note that all of the described PTMs were developed to model colloid transport, but are routinely employed to model ENP transport.16,17,19,31−34 While ENPs may fall within the technical definition of colloids, their physicochemical properties (e.g., composition, shape, isoelectric point, reactivity) may be different.12 However, the mathematical construction of the PTMs does not explicitly consider any physicochemical property of the ENPs, see mass-balance equations in Table 1. None of the parameters in the mass-balance equations reflects type, size, shape, surface properties or aggregation behavior of ENPs. M1 and M2 are the only models where some ENP properties are used. In these models, the parameter η0 is a component of the equation used to estimate the deposition rate constant, kd (Supporting Information eq S5). η0 is quantified according to the Tufenjki−Elimelech correlation equation.27 This equation considers the ENP diameter and van-der-Waals interaction energy of the ENPs with the collector. However, correlation equations for η0 were not developed to consider ENP-soil interactions (e.g., nonspherical particles and collectors, the presence of natural organic matter, steric interactions, and repulsive DLVO interactions) and, as such, are not likely to reflect conditions in ENP column experiments.27,35,36 Furthermore, it is difficult to determine the influence of the ENP properties on the performance of M1 and M2 because η0 is combined with the attachment efficiency, α, in the equation for kd, and α is used as a free parameter whose value is derived from the fit of the model to the RP and/or BTC. Therefore, it may not be η0 that determines the fit of model to the retention profile (and the suitability of the model), but the magnitude of the fitted value of α.

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MATERIALS AND METHODS Particle Transport Models (PTMs). For steady uniform flow vertically through an isotropic homogeneous porous medium, and without a concentration gradient or advection perpendicular to the primary flow direction, the mass balance equation for a substance present in the fluid phase is ρ ∂S ∂C ∂ 2C ∂C = Dp 2 − νp − b ∂t ∂x θ ∂t ∂x

= kdC − k rS

(1) 12734



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Figure 1. Digitized retention profiles (dots) and fitted model results (lines) are presented in panels 1a, 2a, 3a, and 4a; digitized breakthrough curves (dots) and model-derived breakthrough curves (lines) are presented in panels 1b, 2b, 3b, and 4b. Note that the y-axis of panel 1a is presented at a 10fold reduced scale in comparison to panels 2a, 3a, and 4a. Mass balances reported by the original authors for the retained (MR) and the effluent fraction (ME) are shown in panels 1a−4a and panels 1b−4b, respectively.

Table 2. Modeling Parameters for the Retention Profiles in Figure 1 as Reported in the Original Studies retention profile shape parameter

symbol

units

HE

LD

NM

MI

column length inner diameter impulse volume porosity influent concentration pH ionic strength pore velocity particle diameter grain diameter longitudinal dispersivity hamaker constant ENP type location (panel in Figure 1) reference

Lc did Vim θ C0 pH I vp dp dg αL A

[m] [m] [m3] [−] [kg m−3] [−] [mM] [m s−1] [m] [m] [m] [J]

0.12 0.03 1.2 × 10−4 0.46 1 × 10−3 8.5 1.0 2.39 × 10−4 1.90 × 10−7 3.50 × 10−4 6.9 × 10−4 5 × 10−20b MWCNTs 1a and 1b

0.12 0.03 9.0 × 10−5 0.427 1 × 10−2 6−7 1.0 1.17 × 10−5 8.45 × 10−8 5.50 × 10−4 1.1 × 10−4 2.37 × 10−21 Ag 2a and 2b

0.12 0.03 9.0 × 10−5 0.403 1 × 10−2 6−7 5.0 2.94 × 10−4 8.45 × 10−8 5.50 × 10−4 9.7 × 10−4 2.37 × 10−21 Ag 3a and 3b

0.15 0.025 8.7 × 10−5 0.37 2 × 10−2 6 2.25 × 10−2 1.38 × 10−4 1.50 × 10−7 2.75 × 10−4 9.5 × 10−4a 4.5 × 10−20 TiO2 4a and 4b

16

15

15

18

This longitudinal dispersivity was determined by digitizing and fitting the breakthrough curve using CXTFIT. Hamaker constants for MWCNTs were not measured by Kasel et al.16 Instead, a value for single-walled CNTs, which may be considered representative for MWCNTs, was taken from Rajter et al.42 a

41 b

Soil properties that enter the mass-balance equations (Table 1) are bulk density, ρb, porosity, θ, the average diameter of the granular media, dc, and the dispersivity of a tracer in the porous media, Dp (note that Dp is not derived specifically for ENPs). None of the models include parameters to consider the chemical composition, roughness, grain shape, or concentration of natural organic matter of the soil.

The remaining model parameters are rate constants for deposition and remobilization, maximum solid-phase concentration, etc., see Table 1. These parameters are treated as free parameters in the models whose values are determined by fitting the model to the measured RP and/or BTC. In an implicit way, these parameters reflect the properties of the ENPs and of the soil and pore water, and the interaction of the ENPs with the soil under the given conditions, but they are not 12735



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then assess the ability of each model to describe both the retention profile and breakthrough curve. For each model, one or more parameters were estimated using the Levenberg−Marquardt algorithm (LMA), an unbounded, nonlinear, least-squares curve fitting method.37 For several model scenarios, LMA parameter estimation failed to converge or resulted in negative parameters. For these situations, boundaries were applied to the parameter estimation routine using a trust-region nonlinear fitting algorithm.38 Parameter ranges were then chosen to encompass a realistic range of parameter values (i.e., kd, kr, Sm, and β > 0; 0 < α < 1). All parameter estimations employed a multistart global optimization to prevent local minima solutions generated by the influence of initial conditions. Statistical assessment of how well a PTM fits the measured retention profile was performed by evaluation of the biasadjusted Akaike information criterion (AICc).39 Detailed information concerning the AICc analysis is provided in the SI.

determined according to the properties of the ENPs and the system before the model is run. Therefore, the model results, that is, the fitted parameters and the calculated RPs and BTCs, reflect the properties of the ENPs and how ENPs interact with the soil, whereas the models, as they are defined by the massbalance equations, do not take into account any ENP-specific information. Selection of Experimental Data. Data employed in the model assessment were selected as follows. First, more than 100 nanomaterial column transport studies from 2001−2014 were reviewed. Second, 52 of the most recent nanoparticle transport publications (2008−2014) were evaluated in detail to determine whether sufficient information was available to adequately model ENP transport with each PTM; this is summarized in Supporting Information Table S1. Of these 52 publications, 22 (42.3%) reported retention profiles and 12 (23%) publications contained sufficient data to employ all seven PTMs. Of these 12 publications with sufficient data, 6 (11.5%) were excluded because of other limitations (e.g., fewer than 8 retention profile data points; under-saturated soil, and/ or the experimental procedure violates model assumptions of a model employed for this assessment). From the remaining six (11.5%) publications,15−20 a total of 42 retention profiles and associated parameters were recorded and the profiles were modeled using each PTM. Five publications noted HE retention profiles.15−17,19,20 Two publications noted LD retention profiles.19,20 Three publications noted NM retention profiles.17−19 Two publications noted MI retention profiles.17,18 Shape designation was determined by reductive fitting where each profile was fit with linear, exponential, and power functions. The shape designation with the highest adjusted R2 value was selected to describe the profile. A single archetypal example from each retention profile type was then selected for our assessment and is shown in Figure 1; parameters used in the modeling of the four profiles are given in Table 2. A selection of 16 additional HE, LD, NM, and MI retention profiles is provided in the SI (Figure S1). Parameter Fitting. Soil column experiments provide a breakthrough curve (BTC) (i.e., the fraction of material leaving the column, expressed as a function of the number of pore volumes pumped through the column), and a retention profile (i.e., the fraction of ENPs sorbed to the packed bed as a function of depth). The retention profile provides spatial information across the column at the end of the experiment; the breakthrough curve provides temporal information concerning the relative fraction of particles leaving the column. The mechanistic link between the retention profile and breakthrough profile is, in theory, so close that parameter determination from either profile alone should be sufficient to correctly simulate the other. During the model construction phase of this research, we observed that simultaneous fitting of the retention profile and breakthrough curve can result in a suboptimal fit of the retention profile, as the effective sample size (i.e., number of data points vs number of fitted parameters) for fitting the breakthrough curve is often an order of magnitude greater than the effective sample size for fitting the retention profile. The relative weighting of the multiobjective fitting can be manipulated20 and therefore it is unclear what the acceptable goodness-of-fit is. Thus, to best illustrate the conceptual limitations of the models, we estimate model parameters directly from fitting the models to the retention profiles and

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RESULTS AND DISCUSSION For each representative retention profile (HE, LD, NM, and MI) the results and discussion are organized as follows. First, a brief overview of experimental conditions where each retention profile has been observed is provided. Note that this overview contains only a partial list of well-cited and recent publications and should not be considered an exhaustive review. Second, model performance is evaluated with reference to any visually identifiable systematic deviations between the observed and modeled RPs and BTCs (Figure 1); significant results from the AICc model plausibility analysis are presented, when applicable, and the most plausible models for each profile are identified. Note that BTC recession is not modeled. Finally, the suitability of the models is discussed with reference to the findings of the performance evaluation. Parameter fitting results for each profile are available in SI Table S2. We do not discuss these results, nor the relationship of the parameters to the data presented in the original papers, for several reasons. First, our assessment focuses on fitting the measured retention profiles to illustrate the conceptual limitations of the models to describe HE, LD, NM, and MI retention profiles. As such, parameter values derived from our fit may be substantially different from published values determined by fitting models to retention profiles and breakthrough curves simultaneously, as is performed by the software package, Hydrus-1D.40 Second, parameter values from the fit of a PTM to a retention profile cannot be directly compared or validated against values of the same parameters derived from other models (i.e., model construction changes the meaning of the parameters).16 Therefore, no comparative model performance conclusions can be drawn from inspection of the parameter results. Finally, throughout this work it is argued that parameters derived from forced application of the PTMs to fit LD, NM, and MI retention profiles should not be generalized. Our evaluation of the applicability of the models is independent of the value of fitted parameters, and the parameter estimation results are not provided in the main text to prevent their misapplication. Hyperexponential Retention. HE retention profiles have been reported for a variety of ENPs including nano fullerene aggregates in various porous media and ionic strengths,17 titanium dioxide nanoparticles at various solution conditions, concentrations, and flow rates,43,44 copper-bearing hydroxyapatite nanoparticles at various ionic strength conditions,14 12736



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cerium oxide nanoparticles at various pHs,45 copper oxide nanoparticles in various concentrations of humic and fulvic acids,22 nano zinc oxide in various solution conditions both with and without humic acid, and in the presence of an Escherichia coli biofilm,13,46,47 multi-walled carbon nanotubes (MWCNTs) in various column lengths, grain sizes, flow rates, and concentrations,16,48 biochar nanoparticles produced at various pyrolysis temperatures,15 and nanosilver at various solution conditions, concentrations, and flow rates.32 A measured HE RP from Kasel et al.16 for the transport of MWCNTs is presented in Figure 1, panel 1a; along with the fitted RPs from the seven models; BTCs generated by each model are shown in Figure 1, panel 1b. Panel 1a allows facile differentiation between models that can simulate HE RPs from those that cannot. M1, M3, and M5, which cannot model HE retention, show exponential retention profiles that underestimate retention from 0.0 to 0.015 m and overestimate retention from 0.015 to 0.12 m. M2, M6, and M7 well describe the RP, and M4 is in between. If only the retention profile is considered, M6 is the most plausible model (Akaike weight = 0.99). Note that M2 and M7 fit the retention profile with similar error as M6 (RSSM6 = 8.72 × 10−15; RSSM2 = 2.29 × 10−14; RSSM7 = 1.94 × 10−14 RSS: residual sum of squares). However, if the BTC is also considered, M2 and M6 become unsuitable, as they are unable to describe the gradual decrease in retention (increase in breakthrough) exhibited in the breakthrough curve. Therefore, M7 is the most suitable model to describe this set of data. Examination of further profiles from Kasel et al.,16 Wang et al.,15 and Wang et al.17 exhibiting varying degrees of hyperexponentiality shows that M7 is frequently needed to describe both the RP and BTC, but is not always the most suitable model. In two additional cases (SI, Figure S2 and Tables S6 and S7), M2 and M6 most suitably describe the RP and also the BTC. This result is important because it shows that there is no model that is appropriate for all conditions and that the inclusion of additional model parameters does not guarantee better model performance. To avoid applying an overparameterized model to HE RPs, we advise that M2, M6, and M7 are evaluated in congress and an AICc model suitability analysis is employed to select the most plausible model. In using M7, particular attention needs to be paid to the treatment of the parameters β and Sm. It is common practice to predefine β.15,19,20,49 However, this can easily result in suboptimal fit, as shown by the dashed lines in Figure 1; panel 1a and 1b, which represent the RP and BTC of M7 with β predefined to be 0.765, as done by Kasel et al.16 Our result for M7, in contrast, (orange line) shows that leaving β a free parameter gives a superior fit of the RP without introducing much error in the BTC. In addition to suboptimal fitting caused by predefining β, one must also consider that the relationship of β to physical parameters (e.g., grain size, roughness, particle shape) is not currently known. Thus, because fitting is the only methodology to estimate β, there is no justification to predefine the parameter. Regarding Sm, it is possible to determine Sm with batch adsorption experiments, as done by Wang et al.17 This is a good way of supporting the assumption of blocking with independent empirical data. However, the methodology of measuring nanoparticle adsorption must be carefully selected, as nanoparticles do not have partition coefficients like chemicals (i.e., thermodynamic equilibrium), but their sorption is kinetically controlled.50

Linearly Decreasing Retention. LD retention profiles have been reported for a variety of nanomaterials including biochar nanoparticles in iron oxyhydroxide-coated sand and in various concentrations of humic acid,20 nanosilver at low flow rates,19 fullerenes and titanium dioxide nanoparticles at various solution conditions,44 and capped zinc oxide nanoparticles.51 A measured LD retention profile from Liang et al.19 for the transport of nanosilver is presented in Figure 1, panel 2a; along with fitted RPs for all seven models; the corresponding BTCs are shown in Figure 1, panel 2b. M1 and M2 present identical exponential RPs that slightly overestimate retention from 0.0 to 0.01 m, slightly underestimate retention from 0.02 to 0.06 m, and again slightly overestimate retention from 0.08 to 0.12 m. M3−M7 exhibit similar, shallowly sigmoidal-shaped retention profiles that well describe retention from 0.0 to 0.09 m, but slightly overestimate retention from 0.09 to 0.12 m. If only the retention profile is considered, M7 is the most plausible model (Akaike weight = 0.891; RSSM7 = 3.69 × 10−13). According to Figure 1, panel 2b, breakthrough occurred near one pore volume and was proceeded by a gradual decrease in retention (linearly increasing portion of the measured breakthrough curve). Modeled BTCs generated by M3 and M5 more closely describe the gradual decrease in retention exhibited in the measured BTC than M7 does. Detailed inspection indicated that the characteristic gradual decrease in retention typical of blocking effects can, under some circumstances, be described by less complex models (such as M3) when deposition and remobilization rate constants are similar in magnitude. Overall, M3, M5 and M7 have a similar performance in describing the RP and BTC; M3 then is the most plausible model because it has the lowest number of parameters. Concerning the application of M7, this type of profile also provides evidence that predefining β can result in suboptimal fit, as shown by the dashed line (Figure 1; panel 2a and 2b) where β was predefined to be 0.432 by Liang et al.19 Leaving β a free parameter resulted in a superior RP fit without introducing much error in the BTC. The causation for LD retention is not explicitly known, although it may be that LD retention is an interim state between a hyperexponential, or exponential, and uniform retention profile that occurs as a function of blocking and straining. Our assessment results cannot substantiate this theory, however, as we have no information on the evolution of the retention profile over the length of the experiment. As such, the contribution of blocking and straining to the formation of LD retention profiles is not clear. Concerning the development of more suitable models, modifying the PTM to employ zero, or near zero, order deposition kinetics can generate LD retention profiles. However, the degree to which this modification can be reconciled with current theory has not yet been demonstrated and further research is required to quantitatively describe this behavior. Significant insight concerning the development of LD retention profiles may be obtained by evaluating the changes in the retention profile as the experiments progress. Nonmonotonic Retention. NM retention profiles have been reported for a variety of nanomaterials including nano titanium dioxide at various solution conditions, flow rates, and in iron oxide-coated quartz sands both with and without humic acid,18,21,52 fullerene nanoparticles at various flow rates and grain sizes,33 copper oxide nanoparticles in certain organic buffers and solution conditions,22 and nanosilver at various flow 12737



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velocity and the presence of surfactants, and may be caused by the migration of deposed ENPs, competition for retention sites, and/or due to particle aggregate reconformation (i.e., a shift in particle aggregate structure due to aggregate-grain collision).18,19,21,53−55 However, further research is needed to better understand the influence of hydrodynamics, surfactants, and aggregate reconformation on the retention profile, and to implement these mathematically into PTMs. Concerning the development of more suitable models, NM retention profiles can be generated with minimal model adjustment, for instance by including a mobile solid-phase or by allowing β < 0.56 However, how these changes can be experimentally confirmed, or the degree to which these modifications can be reconciled with existing theory for ENPs, has not yet been fully investigated. Nonmonotonic retention profiles can also be simulated by the 2-species model presented by Bradford et al.55 Within the model by Bradford et al.,55 NM retention behavior is mechanistically described using several kinetic exchanges rates (i.e., kinetic exchange rates between aggregated and monodisperse species, and the aggregated species production function). We were not able to employ this model here because the aggregated species production function employed by Bradford et al.55 can only be determined if the aggregated species is tracked in the effluent. This information is not available in any of the six publications with sufficient information about experimental conditions identified in our literature review (see Selection of Experimental Data). Monotonically Increasing Retention. MI profiles are not common, but have been observed for nano titanium dioxide at low ionic strength conditions and in iron oxide-coated quartz sands with humic acid,18,52 and Fullerene nanoparticles at various flow rates and grain sizes.33 A measured MI retention profile reported by Chen et al.18 for the transport of titanium dioxide nanoparticles is presented in Figure 1, panel 4a, along with the fitted RPs for all seven PTMs. In the literature, no applications of any of the seven PTMs to this type of retention profile have been reported. We include this profile because it shows the diversity of results obtained in column experiments with ENPs. For all seven models, the fitted RPs in Figure 1, panel 4a, deviate considerably from the measured retention profile. Consequently, Akaike weights cannot provide reliable information on model plausibility. Figure 1, panel 4b, shows the measured and the modeled BTCs. M1 and M2 yield BTCs that are close to the measured data, but for all other models, the BTCs are very different from the measured data. This example shows that even in cases where the models are obviously inappropriate, agreement between measured and modeled BTCs may be obtained. This adds emphasis to our earlier recommendation that the models should not be used to describe measured BTCs without a detailed investigation of model performance for RPs and BTCs separately. In the study where the presented MI retention profile was observed, Chen et al.18 proposed that observations of increasing retention may be influenced by aggregate reconformation due to collisions between nanoparticle aggregates and sand grains during transport. However, until a PTM that includes aggregate reconformation has been developed, or the understanding of MI retention is more complete, parameters determined from the application of current PTMs to MI retention profiles are meaningless and should not be employed to determine ENP transport.

rates and solution conditions both with, and without, surfactant.19 A measured NM retention profile from Liang et al.19 for the transport of silver nanoparticles is presented in Figure 1, panel 3a; peak retention is observed at ca. 0.012 m from the column inlet. The same graph also shows the fitted RPs for all seven models. Inspection of panel 3a shows two distinct groupings of models. M1 and M2, which do not cover remobilization or dispersion, present exponential profiles that strongly overestimate retention from 0.0 to 0.012 m, underestimate retention from 0.012 to 0.075 m, and well describe retention from 0.075 to 0.12 m. M3−M7, which consider remobilization and dispersion, present essentially identical sigmoidal retention profiles that well describe the near-influent ENP concentration, but clearly underestimate peak retention and poorly describe retention at depth. However, as all of the models show easily identifiable and systematic deviations, none of the models well describes the NM retention profile; it is known that none of the models can explicitly describe NM retention.19 If the models are considered as purely descriptive tools that provide an approximation of the observed RP, M3 is the most plausible model (Akaike weight = 0.761; RSSM3 = 2.12 × 10−11). Inspection of the measured BTC (Figure 1, panel 3b) indicates that a majority of the ENPs were retained throughout the course of the experiment (C/C0 does not exceed 0.05 in Figure 1, panel 3b). Interestingly, the BTCs generated by M3 as the most plausible model, but also by M4−M7, are qualitatively different from the observed BTC because they show no breakthrough at all. The BTCs generated by M1 and M2, in contrast, are in perfect agreement with the measured BTC. Importantly, when we allow a higher error tolerance (HET) in the parameter fitting routine for the RP, the BTCs obtained with M3−M7 change (shown as the dashed line in Figure 1, panels 3a and 3b) and agree very well with the measured BTC, and also converge with the BTCs of M1 and M2. Under these conditions M1 is the most plausible model, because it describes both RP and BTC equally well as all other models and has the lowest number of parameters. This case, where model results are in poor agreement with the measured RP, demonstrates an important limitation of using these PTMs in a parameter fitting exercise for measured RPs and BTCs. For the more complex models, M3−M7, the best fit for the RP leads to a poor fit of the BTC, and vice versa. The conceptual problem in this situation is that there is no agreed-upon acceptable fitting error, which makes the fitting exercise highly subjective. Simultaneous fit of the models to both RP and BTC provides no solution here, because this just masks the lack of model performance on one of the two subsets of the data, RP or BTC. Comparing the performance of all seven PTMs shows, in hindsight, that the conceptual problem can, in this particular case, be avoided by using M1 or M2. However, this is a purely ad-hoc solution and illustrates that the most plausible model can only be found by trial and error and that the models are not based on a sufficient understanding of the processes governing the fate of ENPs in the column. Finally, this case shows that simultaneous fitting of the RP and BTC may mask the inability of a model to describe the RP. Currently, the causation of NM retention behavior is not explicitly known. Experimental evidence suggests that NM retention profiles are qualitatively influenced by the pore water 12738



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the ratio of particle diameter to collector diameter, dp/dc;58 heterogeneity in surface charge characteristics can be evaluated by inspection of the variability of collector size/surface charges throughout the column or by observations of dependence, or independence, of grain size on the magnitude of retention;55 and direct inspection of particle remobilization may help distinguish wedging from straining mechanisms.23,24 These metrics may lead to more robust models in the future, but insufficient mechanistic understanding still prevents these metrics from yielding unequivocal process identification currently. For example, a dp/dc ratio below 8 × 10−3 implies that straining is not likely to be responsible for depthdependent retention, but it does not definitively indicate which mechanism is responsible.58 As depth-dependent retention may also be caused by mechanisms other than straining, such as heterogeneity in particle attachment efficiencies, wedging, and variability of colloid and collector charge characteristics,5,59 this metric provides only qualitative information about the plausibility of straining. Our analysis shows that the applicability of current PTMs to RPs and BTCs measured for different ENPs varies widely. A first important recommendation for the selection of appropriate models is that PTMs should not directly be fitted to RPs and BTCs simultaneously. Instead, model performance should be investigated separately for RPs and BTCs. Second, models should be employed only under conditions where sufficient parameter validation is possible to justify the selection of a particular model. Without sufficient parameter validation, the application of these models, even application that results in a high goodness-of-fit, is not sufficient to infer which processes actually occur within the soil column. For example, in several cases, BTCs can be described relatively well even when a model is obviously not able to reproduce the corresponding RP (see MI profile and BTC in Figure 1, panels 4a and 4b). However, such a result is purely descriptive, does not provide any mechanistic insights, and cannot be generalized. This does not imply that the current models are invalid and cannot, or should not, be used. To the contrary, we argue that these models should be employed, but only under conditions where sufficient parameter validation is possible to justify the selection of a particular model. We recommend that model practitioners avoid forced application of fundamentally unsuitable models, as well as refrain from modeling practices that may inadvertently misconstrue or misrepresent transport parameters and processes.

ENVIRONMENTAL IMPLICATIONS Soil column experiments conducted with ENPs exhibit a much wider range of retention profiles than can be described by current PTMs; the observed RPs are qualitatively very diverse and include HE but also LD, NM and MI shapes. HE RPs can be described by some of the presented PTMs, but NM, LD, and MI retention profiles cannot, mathematically, be generated by any of these models. Several important implications are noted concerning the applicability of the presented PTMs to simulate HE, NM, LD, and MI retention profiles from soil column experiments with ENPs. Hyperexponential Profiles. Models M2, M4, M6, and M7 can generate HE RPs; of these, M2, M6 and M7 show good fits of the RP in Figure 1, panel 1a. With the parameters obtained from the fit of the RP, M7 also describes the BTC with good agreement (Figure 1, panel 1b). However, we do not recommend M7 as the most plausible PTM for all HE RPs and suggest that multiple models (i.e., M2, M6, and M7) be applied in congress and the best model selected using an AICc evaluation. Regarding the common practice of predefining β, we recommend that without experimental verification that blocking actually occurs within the column, or until a clear physical meaning for β becomes apparent, β should remain a free parameter. Other Profiles. The other profiles cannot be mathematically generated by any of the PTMs considered here. In some cases, some of the models have satisfying descriptive performance, for example M3, M5, and M7 for the LD profile and the corresponding BTC. However, because the models do not capture the underlying mechanisms for these RPs, they have no predictive power and it is not clear to what extent good descriptive power of a model that was observed in a specific case can be generalized. In particular, forced application of PTMs to NM (and MI) retention profiles results in large and/ or systematic deviations between models and measured RPs, and poor model performance. Therefore, parameters determined from the application of existing PTMs to NM and MI profiles are meaningless and should not be used for ENP fate assessment. Concerning the application of PTMs to simulate LD retention profiles, the situation is more complex. In contrast to NM and MI retention profiles, LD retention profiles can be approximated by the PTMs presented here. However, because the PTMs cannot explicitly generate LD retention, parameters determined from a fit of these PTMs to LD retention profiles have no predictive power and are inappropriate for ENP fate assessment. Blocking. We observe that the effect of a term for blocking, such as in M7, can also be accounted for by models that do not explicitly include a term for blocking, such as M3, under some circumstances. This implies that it may not be possible to infer whether or not blocking occurs within the column solely by model application to RPs and BTCs. To prevent model misapplication and unjustified inferences on processes taking place in the column, it is recommended to employ a PTM that includes blocking only after experimental verification that blocking occurs within the column, and after independent determination of the parameter Sm.57 Model Selection. Several experimental and theoretical metrics can be employed to enhance model selection and gain process understanding. Factors leading to nonexponential behavior can be evaluated by inspection of experimental conditions. For example, straining may be evaluated by using

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ASSOCIATED CONTENT

S Supporting Information *

Supporting model and method information, and simulation and plausibility results are provided in the SI. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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